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An orthogonal trajectory intersects every regular member of a specified family of plane curves at a right angle. If eliminating the family parameter gives its tangent slope $dy/dx=F(x,y)$, the dot product of the two tangent directions gives the orthogonal slope $dy/dx=-1/F(x,y)$ wherever both slopes are finite and nonzero. Vertical and horizontal tangent directions are interpreted geometrically. For the family $y=c{x}^n$ with $n>0$, eliminating $c$ gives $F=ny/x$, and the orthogonal trajectories satisfy $x^2+n{y}^2=C$. For $n=2$ these are ellipses; for $n=1$ they are circles.

Suppose $h$ is bounded by one in absolute value on a punctured neighbourhood of zero. Suppose also that there are positive sequences $p_n,q_n\to 0$ with $h(p_n)=1$ and $h(q_n)=0$. Extend $F_a(x)=|x{|}^{a}h(x)$ by $F_a(0)=0$. ThenWhen the derivative exists, it equals zero. For continuity, the bound $|F_a(x)|\le |x{|}^a$ gives the epsilon-delta definition of continuity with $\delta ={\epsilon }^{1/a}$ when $a>0$. If $a\le 0$, the values $F_a(p_n)=p_n^a$ remain at least one for sufficiently large $n$, contradicting continuity at zero.

For differentiability, the difference quotient is bounded in absolute value by $|x{|}^{a-1}$, so it tends to zero when $a>1$. Conversely its values along $q_n$ are zero, forcing any possible derivative to be zero. Its values along $p_n$ equal $p_n^{a-1}$ and fail to tend to zero when $a\le 1$.

For example, $h(x)=\sin (\pi \sin (1/x))$ has the required positive sequences $p_n=1/(2\pi n+\pi /6)$ and $q_n=1/(2\pi n)$. This separates the suppression needed for continuity from the stronger suppression needed for differentiability.

Anomers differ only in configuration at the anomeric carbon of the same cyclic monosaccharide form.

The incremental capacitance associated with changes of electric charge around a p-n junction. Its depletion contribution is often estimated by $C_j\simeq {\epsilon }_{s}A/W$, so an independent junction capacitance measurement can constrain the width $W$ of the depletion region under the model assumptions. Other stored electric charge, including injected minority carriers, can also contribute to a measured terminal capacitance.

The depletion approximation replaces the junction's transition region by a region without mobile carriers, leaving only fixed ionized dopants. For a uniformly doped abrupt p-n junction, $\rho =-e{N}_A$ on the p side, $\rho =+e{N}_D$ on the n side and $\rho \simeq 0$ outside. Charge neutrality gives $N_A{x}_p=N_D{x}_n$. Integrating Poisson equation gives the electric field and the electrostatic band bending.

Thermal generation of electron-hole pairs within a reverse-biased depletion region can supply a leakage electric current in addition to neutral-region diffusion. For a simple midgap trap with equal electron and hole lifetime parameters, its scale is $e\mathcal{A}W{n}_i/(2{\tau }_g)$, where $W$ is depletion width and ${\tau }_g$ the lifetime parameter. The electric field separates generated carriers. Unlike ideal diode minority-carrier diffusion current, this contribution scales approximately with $n_i$ rather than $n_i^2$ and can vary with reverse voltage through $W$. Consequently a measured activation energy need not equal the full band gap.

The Asakura-Oosawa model treats dilute nonadsorbing polymer coils as mutually interpenetrable ideal particles excluded from a layer around each colloid. For colloid radius $a$ and polymer exclusion thickness $R_g$, two exclusion spheres of radius $R_e=a+R_g$ overlap when $2a\le R<2R_e$. Adding their two spherical cap volumes gives $V_{\mathrm{ov}}=\pi (4R_e+R)(2R_e-R{)}^2/12$. Multiplication by $-{\Pi }_p$ gives the displayed pair free energy, with zero depletion interaction beyond $2R_e$. The range is controlled by polymer size and the depth by reservoir polymer concentration. At larger polymer concentration, polymer interactions and many-body overlap effects require care.

A DNA lesion is physical or chemical damage to DNA, such as an altered nucleobase or strand break. DNA repair can restore the sequence before a lasting mutation occurs.

Gas density is the mass density of a gas. For an ideal gas sample of amount of substance $n$, mean molar mass $\overline{M}$ and volume $V$, combine $m=n\overline{M}$ with $pV=nRT$ to obtainHere $R$ is the molar gas constant. For a homogeneous mixture of gases, $\overline{M}={\sum }_i{x}_i{M}_i$, where $x_i$ is the mole fraction of component $i$. Equal pressure and temperature make a gas density ratio equal to a mean-molar mass ratio within this model. Argon detection in a binary gas mixture gives a worked example and a temperature confounder.

For a spherical surface of radius $R$ with exterior mass density $\rho (r)={\rho }_0{e}^{-(r-R)/h_0}$, integrate the spherical volume element to obtain $M=4\pi {\rho }_0{\int }_0^{\infty }(R+h{)}^2{e}^{-h/h_0}dh$. The moments ${\int }_0^{\infty }{h}^n{e}^{-h/h_0}dh=n!h_0^{n+1}$ give the displayed expression. Its leading thin-atmosphere approximation is surface area times column mass, $4\pi R^2{\rho }_0{h}_0$; the two positive correction terms account for the growth of shell area with altitude.

For $H=p^2/(2m)-g\delta (x)$ with $g>0$, put $\kappa =mg/{\hslash }^2$ and $T>0$. The bound state in a delta potential has squared amplitude $\kappa$ at the origin. The even normalized continuum states have squared amplitude $k^2/[\pi (k^2+{\kappa }^2)]$ for $k>0$, and the odd states vanish there. The spectral resolution of a quantum propagator gives the displayed result. At long times, the continuum stationary point at $k=0$ is suppressed by its quadratic spectral weight: expanding that weight gives a continuum term $e^{-3i\pi /4}(2m/\hslash T{)}^{3/2}/(4\sqrt{\pi }{\kappa }^2)+O(T^{-5/2})$. The bound state term persists with constant modulus. This stationary phase expansion requires $\hslash {\kappa }^{2}T/(2m)\gg 1$.

Two distinguishable particles with equal one-particle spectra have degenerate ordered orbital products $|r,s⟩$ and $|s,r⟩$ for $r\ne s$. Suppose both individual position means vanish, as in a centred parity-symmetric well. The perturbation $\alpha x_1{x}_2$ then has zero diagonal matrix elements, but its restriction to this two-dimensional subspace isDegenerate perturbation theory gives normalized symmetric and antisymmetric eigenfunctions $(|r,s⟩\pm |s,r⟩)/\sqrt{2}$ with shifts $\pm \alpha |x_{rs}{|}^2$. Thus vanishing diagonal expectations in a chosen degenerate basis do not imply vanishing first-order energy shifts. Distinguishability permits both exchange symmetries; it does not forbid their energy splitting.

For an infinite square well of width $a$, the first two orbitals have $|x_{12}{|}^2=256a^2/(81{\pi }^4)$ by the position matrix elements in a symmetric infinite square well. If the particles are identical spin one-half fermions, the spatially symmetric and antisymmetric combinations must instead be paired with the singlet state and triplet states, respectively. A spin-independent perturbation leaves each allowed spin multiplicity intact.